Final answer:
The z-score corresponding to an area under the standard normal curve of 0.4314 is approximately -0.17. This is determined using a Z-table which correlates areas to z-scores for a standard normal distribution with mean 0 and standard deviation 1.
Step-by-step explanation:
Calculating The Z-Score
Given that the probability (area under the standard normal curve) to the left of the z-score is 0.4314, we need to find the corresponding z-score, a. This involves using a Z-table which lists the areas under the standard normal curve to the left of different z-scores. A Z-score tells us how many standard deviations an observation is above or below the mean.
To find our z-score, we refer to the Z-table and look for the area closest to 0.4314. We find that the corresponding z-score for an area of 0.4314 is approximately -0.17. Since the standard normal distribution has a mean of 0 and a standard deviation of 1, we use the formula z = (x - μ)/σ, where μ is the mean and σ is the standard deviation, to find that a = -0.17 when rounded to two decimal places.